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Questions and Answers
Simplify the expression: $(2\sqrt{3} + 3\sqrt{2})^2$.
Simplify the expression: $(2\sqrt{3} + 3\sqrt{2})^2$.
48 + 36√6
Simplify the expression: $4^{\frac{2}{3}} × 4^{\frac{1}{3}}$.
Simplify the expression: $4^{\frac{2}{3}} × 4^{\frac{1}{3}}$.
8
Simplify the surd: $5\sqrt{75}$.
Simplify the surd: $5\sqrt{75}$.
25\sqrt{3}
If $a = 2^{\frac{1}{2}}$, $b = 2^{\frac{1}{4}}$, and $c = 2^{\frac{3}{4}}$, find the value of $a^2b^4c^2$.
If $a = 2^{\frac{1}{2}}$, $b = 2^{\frac{1}{4}}$, and $c = 2^{\frac{3}{4}}$, find the value of $a^2b^4c^2$.
Simplify the expression: $\frac{4^{\frac{2}{3}}}{4^{\frac{1}{3}}}$.
Simplify the expression: $\frac{4^{\frac{2}{3}}}{4^{\frac{1}{3}}}$.
Flashcards
Simplify (2√3 + 3√2)^2
Simplify (2√3 + 3√2)^2
Expanding the binomial (2√3 + 3√2)^2 results in 48 + 36√6.
Simplify 4^(2/3) × 4^(1/3)
Simplify 4^(2/3) × 4^(1/3)
Multiplying exponential terms with the same base means adding their exponents: 4^(2/3 + 1/3) = 4^1 = 8.
Simplify 5√75
Simplify 5√75
Simplifying 5√75 results in 25√3. This involves simplifying the square root of 75, recognizing 75 as 25 * 3.
If a = 2^(1/2), b = 2^(1/4), c = 2^(3/4), find a^2b^4c^2.
If a = 2^(1/2), b = 2^(1/4), c = 2^(3/4), find a^2b^4c^2.
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Simplify 4^(2/3) / 4^(1/3)
Simplify 4^(2/3) / 4^(1/3)
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Study Notes
Surds and Indices
Surds
- A surd is a value that, when multiplied by itself, gives a whole number (e.g. √2, √3, etc.)
- Surds can be simplified by finding the largest perfect square that divides the number under the square root, and then taking that square root out (e.g. √12 = √(4 × 3) = 2√3)
- Rationalising a surd involves eliminating the surd from the denominator of a fraction by multiplying both numerator and denominator by the surd (e.g. 1/√2 = √2/2)
- Surds can be added and subtracted when they have the same base and index (e.g. 2√3 + 3√3 = 5√3)
- Surds can be multiplied and divided by using the property √a × √b = √(a × b) and √a ÷ √b = √(a ÷ b)
- When adding and subtracting surds with different bases, convert to the same base by finding a common multiple of the bases (e.g. √2 + √8 = √2 + √(4 × 2) = √2 + 2√2)
Indices
- Indices, also known as powers or exponents, are used to express very large or very small numbers in a more compact form (e.g. 2^3 means 2 to the power of 3, or 2 × 2 × 2)
- The index laws state that a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m-n), and (a^m)^n = a^(m×n)
- Indices can be used to simplify fractions by using the index laws (e.g. 2^3/2^2 = 2^(3-2) = 2^1 = 2)
- When simplifying fractions with indices, the numerator and denominator must have the same base, and the indices can be subtracted (e.g. 2^5/2^3 = 2^(5-3) = 2^2 = 4)
Note: These study notes provide a concise summary of the key concepts and formulas related to surds and indices at the 10th grade level in Australia. They are meant to serve as a quick reference guide for students to study and review these topics.
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